### Rotating Fourier Series

**Chapter 3 Summation of rotating vectors exp(jωt)**

**Chapter 3.1 What’s that for?**To make it easier to understand the summation of the

**cos(ωt)**function, more broadly –

**sinusoidal**type functions.

And how he found it fits

**Fourier Series**, which is the sum of

**sines**and

**cosines**.

**Chapter 3.2 General summation of rotating vectors exp(jωt)**

**Fig.3-1**

Fig.3-1a

Sum of rotating vectors

Fig.3-1a

**F(jt)**where:

**c0**is a constant component or, in other words, a rotating vector with a speed

**ω=0**

**c1, c2, c3…**is a complex number that is the

**initial state**of each spinning vector

**c1*exp(jω1t), c2exp(jω2t),ec3xp(jω3t)…**

**ω1, ω2, ω3…**The pulsations of

**ω1, ω2, ω3…**are

**arbitrary**! i.e. may or may not be

**multiples**of the first harmonic

**ω1**. This is the most general

**sum**formula and we will not analyze this case.

**Fig.3-1b**

Sum of rotating

**F(jnω0t)**vectors with the same

**ω0**pulsation.

It is a

**complex number**vector

**c=c1+c2+c3+…**rotating at a speed of

**ω0**.

The

**real**number

**c0**is a constant component.

The

**sum**of

**sine**and

**cosine**is also a

**sine**with the same pulsation

**ω0**and the corresponding

**phase**shift

**φ**.

It is used in classical

**electrical**engineering, where power plants generate electricity with a

**constant**frequency

**f=50Hz,**i.e.

**ω0=314*1/sec**. The topic has already been thoroughly worked out, probably in the

**19th**century.

Example with animation in

**chapter 3.3**

**Fig.3-1c**

This is what we’ll mainly be dealing with.

i.e. sum of rotating vectors

**F(njω0t)**with increasing pulsations

**1ω0,2ω0, 3ω0…**

I emphasize that each

**pulsation**is a

**multiple**of the basic pulsation

**ω0**, and not any

**ω1, ω2, ω3 …**as in

**Fig. 3-1a**

Examples with animation in

**chapters 3.4…7**

**Chapter 3.3 Sum of 2 rotating vectors exp(jωt) with the same pulsation ω0=1/sec i.e. ****1*exp(j1t)-j1exp(j1t)**

The case of **Fig.3-1b** when **c0=0, c1=1, c2=1*exp(-jπ/2)=-j1 **and** c3=c1+c2=1-j1=√2*exp(-jπ/4 )**

The numbers** c1, c2** and **c3** are the initial states of the rotating vectors as in **Fig.3-2** before the animation.

Notice that the vector **c2** is lagging behind by **π/2=90º** with respect to **c1**.

**Fig. 3-2**Sum of

**2**rotating vectors as:

**1exp(j1t)-j1*exp(j1t)**or

**(1-1j)exp(j1t)**or

**√2exp(-jπ/4)*exp(j1t)**where

**-π/4=-45º**. The

**right**vector is at any time the sum of the

**2 left**vectors and has parameters

**A=√2**,

**ω=1/sec**and

**ϕ=-π/4=-45º**. At the initial moment, i.e. before pressing

**“Start”**, the sum, i.e. the right vector, agrees (vectorically add the left vectors–>”diagonal of the square”). Then pause the simulation at any time by clicking on the drawing or

**“Start”**and check if it is correct.

**The most important conclusion.**

When all vectors have the same rotational speed ω0, then their sum is also a rotating vector with the same speed ω0, length A which is constant and phase ϕ.

**Note1**

Applies to any number of rotating vectors.

**Note2**

Notice that the relative position of all

**3**rotating

**vectors**is

**constant**! This makes it very easy to analyze e.g. electrical circuits. When this is not the case, e.g.

**Fig. 3-1a**or

**3-1c**, and the vectors rotate at different

**speeds**, the sum vector changes

**length**and

**speed**! You will find out in

**Chapter 3.4**.

**Chapter 3.4 Sum F(njω0t)=1exp(j1t)+1exp(j2t)****Chapter 3.4.1 “Vector Only” Version**

We begin to study sums of rotating vectors with different pulsations. “Vector only” means that the ends of the vectors do not draw a trajectory.

The case of** Fig.3-1c** when **c0=0 c1=1, c2=1** and** ω0=1/sec**

**Fig. 3-3****F(jω0t)=1*exp(1jt)+1*exp(2jt) (ω0=1/sec)**

You see vector functions as rotating vectors**1*exp(1jt), 1*exp(2jt)** and their sum** 1*exp(1jt)+1*exp(2jt)**

You can clearly see **2** times the speed of **1*exp(2jt)**. Try to stop the animation at different times **t** and check if the right function is a **vector** sum of the left **2**. Here

**Chapter 3.4.2 Trajectory or “Trace only” version**

“Trace only” means that the ends of the vectors draw a trace. The vectors themselves are invisible and this trace is the trajectory **F(jω0t)**. This is the case of **Fig.3-1c** when **c0=0 c1=1, c2=1** and **ω0=1/sec**. Only one animation period is shown. Further rotations follow the same paths and the animation looks static! Also in the next animations.

Here and further we will limit ourselves to the “only with a trace” version. The “vector” version of this case is the animation in **Fig 3-3**.

**Fig. 3-4**

**F(njω0t)=1*exp(1jt)+1*exp(2jt)**

The right animation is the sum of the

**2**left ones. Note that

**1*exp(2jt)**“stopped” after the first

**half-period**. But the trajectory continues to rotate, only along the same path!

**Chapter 3.5 F(njω0t)= 1exp(j1t)+0.7exp(j2t)**

**Fig. 3-5**

**F(jω0t)=**

**1exp(j1t)+0.7exp(j2t)**

**Chapter 3.6 F(njω0t)=1exp(j1t)+1exp(j2t-π/6)**

**Fig. 3-6**

**F(jω0t)=1exp(j1t)+1exp(j2t-π/6)**

**φ= -π/6**is a decent delay

**φ=-30º**.

The previous examples had a phase shift

**φ=0º**, now one of the components

**1exp(j2t-π/6)**has φ

**non-zero**. The “delaying” component

**1exp(j2t-π/6)**causes the trajectory to rotate to the left.

**Chapter 3.7 F(njω0t)=0.3exp(j1t)+0.5exp(j2t-π/6)+0.45exp(j2t+π/4)**

A more complicated trajectory.

**Fig. 3-7**

**F(njω0t)=0.3exp(j1t)+0.5exp(j2t-π/6)+0.45exp(j2t+π/4)**

Recall that the sum vector keeps rotating, even after the animation ends after the first period

**T**. The formula in

**Fig. 3-7**describes the trajectory for

**ω0=1/sec**. The shape of the trajectory will be exactly the same for

**ω0=2/sec, ω0=3/sec…**Except that it will rotate

**2, 3…**times faster.

The

**red**point in

**(0,0)**is also the so-called

**center of gravity**of the trajectory. More on this in the next chapter.

**Chapter 3.8 Conclusions**

**Fig. 3-8**

F(njω0t)as a

F(njω0t)

**sum**of rotating vectors in which:

–

**c0**is a constant component

**or**formally a rotating vector with

**ω=0**pulsation. Is a real number (and complex at the same time!)

**– c1exp(1jω0)**is a rotating vector with a speed of

**1ω0**

**– c2exp(2jω0)**is a rotating vector with a speed of

**2ω0**

**– c3exp(3jω0)**is a rotating vector with speed

**3ω0**

…

**And**the complex numbers

**c1, c2, c3…**are the

**initial**states of these rotating vectors.

Some treat the initial state as the beginning of the world, and others as the moment

**t=0,**when we start the experiment. Read->press animation. Take a quick look at

**Fig. 4-8**in the next section. There are coefficients

**c0=-0.5, c1=0.9-j0.6, c2=0.6+j0.4**and

**c3=0.4-j0.2**for a particular

**Fourier Series**. The trajectory

**F(njω0t)**is drawn every period

**T**along the same path. This period corresponds to the

**first**harmonic here

**1ω0=1/sec**and

**T≈6.28sec**.